Lineare Differentialgleichung/Konstant/-1324/2-7/Picard-Lindelöf/Bis dritte Iteration/Aufgabe/Lösung

Die nullte Iteration ist die konstante Funktion

{}\varphi _{0}(t)={\begin{pmatrix}x_{0}(t)\\y_{0}(t)\end{pmatrix}}={\begin{pmatrix}2\\-7\end{pmatrix}}\,.

Die erste Iteration ist

{}{\begin{aligned}\varphi _{1}(t)&={\begin{pmatrix}x_{1}(t)\\y_{1}(t)\end{pmatrix}}\\&={\begin{pmatrix}2\\-7\end{pmatrix}}+\int _{0}^{t}{\begin{pmatrix}-1&3\\2&4\end{pmatrix}}{\begin{pmatrix}2\\-7\end{pmatrix}}ds\\&={\begin{pmatrix}2\\-7\end{pmatrix}}+\int _{0}^{t}{\begin{pmatrix}-23\\-24\end{pmatrix}}ds\\&={\begin{pmatrix}2-23t\\-7-24t\end{pmatrix}}.\end{aligned}}

Die zweite Iteration ist

{}{\begin{aligned}\varphi _{2}(t)&={\begin{pmatrix}x_{2}(t)\\y_{2}(t)\end{pmatrix}}\\&={\begin{pmatrix}2\\-7\end{pmatrix}}+\int _{0}^{t}{\begin{pmatrix}-1&3\\2&4\end{pmatrix}}{\begin{pmatrix}2-23s\\-7-24s\end{pmatrix}}ds\\&={\begin{pmatrix}2\\-7\end{pmatrix}}+\int _{0}^{t}{\begin{pmatrix}-23-49s\\-24-142s\end{pmatrix}}ds\\&={\begin{pmatrix}2-23t-{\frac {49}{2}}t^{2}\\-7-24t-71t^{2}\end{pmatrix}}.\end{aligned}}

Die dritte Iteration ist

{}{\begin{aligned}\varphi _{3}(t)&={\begin{pmatrix}x_{3}(t)\\y_{3}(t)\end{pmatrix}}\\&={\begin{pmatrix}2\\-7\end{pmatrix}}+\int _{0}^{t}{\begin{pmatrix}-1&3\\2&4\end{pmatrix}}{\begin{pmatrix}2-23s-{\frac {49}{2}}s^{2}\\-7-24s-71s^{2}\end{pmatrix}}ds\\&={\begin{pmatrix}2\\-7\end{pmatrix}}+\int _{0}^{t}{\begin{pmatrix}-23-49s-{\frac {277}{2}}s^{2}\\-24-142s-333s^{2}\end{pmatrix}}ds\\&={\begin{pmatrix}2-23t-{\frac {49}{2}}t^{2}+{\frac {277}{6}}t^{3}\\-7-24t-71t^{2}-111t^{3}\end{pmatrix}}.\end{aligned}}